#456779

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,31)(29,32);(1,3,4,2)(10,13)(14,16)(17,19,21,30)(18,23,24,27)(20,29,22,28)(25,31,26,32);(2,3)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(20,22)(23,31)(27,32)(29,30);(6,8)(11,15)(14,16)(17,20,21,22)(18,25,24,26)(19,29,30,28)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=ei,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363975);